8(7n-8)-10n(-3n+9)=1

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Solution for 8(7n-8)-10n(-3n+9)=1 equation:


Simplifying
8(7n + -8) + -10n(-3n + 9) = 1

Reorder the terms:
8(-8 + 7n) + -10n(-3n + 9) = 1
(-8 * 8 + 7n * 8) + -10n(-3n + 9) = 1
(-64 + 56n) + -10n(-3n + 9) = 1

Reorder the terms:
-64 + 56n + -10n(9 + -3n) = 1
-64 + 56n + (9 * -10n + -3n * -10n) = 1
-64 + 56n + (-90n + 30n2) = 1

Combine like terms: 56n + -90n = -34n
-64 + -34n + 30n2 = 1

Solving
-64 + -34n + 30n2 = 1

Solving for variable 'n'.

Reorder the terms:
-64 + -1 + -34n + 30n2 = 1 + -1

Combine like terms: -64 + -1 = -65
-65 + -34n + 30n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-65 + -34n + 30n2 = 0

Begin completing the square.  Divide all terms by
30 the coefficient of the squared term: 

Divide each side by '30'.
-2.166666667 + -1.133333333n + n2 = 0.0

Move the constant term to the right:

Add '2.166666667' to each side of the equation.
-2.166666667 + -1.133333333n + 2.166666667 + n2 = 0.0 + 2.166666667

Reorder the terms:
-2.166666667 + 2.166666667 + -1.133333333n + n2 = 0.0 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + -1.133333333n + n2 = 0.0 + 2.166666667
-1.133333333n + n2 = 0.0 + 2.166666667

Combine like terms: 0.0 + 2.166666667 = 2.166666667
-1.133333333n + n2 = 2.166666667

The n term is -1.133333333n.  Take half its coefficient (-0.5666666665).
Square it (0.3211111109) and add it to both sides.

Add '0.3211111109' to each side of the equation.
-1.133333333n + 0.3211111109 + n2 = 2.166666667 + 0.3211111109

Reorder the terms:
0.3211111109 + -1.133333333n + n2 = 2.166666667 + 0.3211111109

Combine like terms: 2.166666667 + 0.3211111109 = 2.4877777779
0.3211111109 + -1.133333333n + n2 = 2.4877777779

Factor a perfect square on the left side:
(n + -0.5666666665)(n + -0.5666666665) = 2.4877777779

Calculate the square root of the right side: 1.577269089

Break this problem into two subproblems by setting 
(n + -0.5666666665) equal to 1.577269089 and -1.577269089.


Subproblem 1
n + -0.5666666665 = 1.577269089

Simplifying
n + -0.5666666665 = 1.577269089

Reorder the terms:
-0.5666666665 + n = 1.577269089

Solving
-0.5666666665 + n = 1.577269089

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '0.5666666665' to each side of the equation.
-0.5666666665 + 0.5666666665 + n = 1.577269089 + 0.5666666665

Combine like terms: -0.5666666665 + 0.5666666665 = 0.0000000000
0.0000000000 + n = 1.577269089 + 0.5666666665
n = 1.577269089 + 0.5666666665

Combine like terms: 1.577269089 + 0.5666666665 = 2.1439357555
n = 2.1439357555

Simplifying
n = 2.1439357555


Subproblem 2
n + -0.5666666665 = -1.577269089

Simplifying
n + -0.5666666665 = -1.577269089

Reorder the terms:
-0.5666666665 + n = -1.577269089

Solving
-0.5666666665 + n = -1.577269089

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '0.5666666665' to each side of the equation.
-0.5666666665 + 0.5666666665 + n = -1.577269089 + 0.5666666665

Combine like terms: -0.5666666665 + 0.5666666665 = 0.0000000000
0.0000000000 + n = -1.577269089 + 0.5666666665
n = -1.577269089 + 0.5666666665

Combine like terms: -1.577269089 + 0.5666666665 = -1.0106024225
n = -1.0106024225

Simplifying
n = -1.0106024225


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.1439357555, -1.0106024225}

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